Why does the hot water work but the cold is slow from the kitchen faucet?

It is probably due to a blockage somewhere in the coldwater pipe or faucet. The easiest to check is the faucet. Turn off the water supply valves. Then take apart the faucet and look for tiny pieces of material (usually metal) that might block the flow of water thru the portholes. If there is no blockage, then the next probable cause is blockage in the pipe. This can occur in copper pipes anywhere there is a soldered joint. The blockage comes from using too much solder so it builds up in the pipe. It is most likely the faucet. Even if you don't see blockage it could be there. You can usually replace the faucet "guts" without too much expense and its worth a try, before working on the water pipes.